I have a context free grammar (CFG). The initial symbol is `A1`. The terminal symbols are just integers and the non-terminal symbols start with `A`. I also have a target sequence of terminal symbols which needs to be produced by the grammar. However, the CFG production rules have had some terminal symbols removed which may now render it impossible to produce the target sequence. Please attempt to repair the CFG so that the sequence can be produced. The only operations which can be made to repair the CFG are the insertion of terminal symbols anywhere in existing production rules. Deleting or reordering symbols in existing production rules and creating new production rules is prohibited. Please try minimize the total number of terminal symbols inserted into the production rules so that the CFG can produce the target sequence. Reason about the problem however you wish and write out whatever working out you want. When you are finished please write a new line with `Updated production rules:`, and then write a new line for each of the updated production rules (including `rule_<n>:`), no need to rewrite the production rules which haven't changed. Do not write anything after the updated production rules.


"""
There are 41 terminals, ranging from 1 to 41.
Each terminal symbol occurs in the target sequence at least once.

There are 63 non-terminal symbols, ranging from A1 to A63.

Production rules:
rule_1: A1 -> A2
rule_2: A2 -> A63
rule_3: A3 -> A4 A5
rule_4: A4 -> 8
rule_5: A5 -> 5 A6
rule_6: A6 -> 1
rule_7: A6 -> 2
rule_8: A6 -> 9
rule_9: A7 -> 10
rule_10: A8 -> A9 A10
rule_11: A9 -> 11
rule_12: A10 -> 3 A12
rule_13: A10 -> 4 A11
rule_14: A11 -> 1
rule_15: A11 -> 2
rule_16: A12 -> 1
rule_17: A12 -> 2
rule_18: A13 -> A14 A15
rule_19: A14 -> 12
rule_20: A15 -> 3 A16
rule_21: A15 -> 4 A17
rule_22: A16 -> 1
rule_23: A16 -> 2
rule_24: A17 -> 1
rule_25: A17 -> 2
rule_26: A17 -> 13
rule_27: A18 -> A19 A20
rule_28: A19 -> 17
rule_29: A20 -> 14 A21
rule_30: A20 -> 15 A22
rule_31: A20 -> 16 A23
rule_32: A21 -> 1
rule_33: A21 -> 2
rule_34: A22 -> 1
rule_35: A22 -> 2
rule_36: A23 -> 1
rule_37: A23 -> 2
rule_38: A24 -> 18
rule_39: A25 -> 14 A26
rule_40: A25 -> 15 A27
rule_41: A25 -> 16 A28
rule_42: A26 -> 1
rule_43: A26 -> 2
rule_44: A26 -> 19
rule_45: A27 -> 1
rule_46: A27 -> 2
rule_47: A28 -> 1
rule_48: A28 -> 2
rule_49: A29 -> 14 A30
rule_50: A29 -> 15 A31
rule_51: A29 -> 16 A32
rule_52: A30 -> 1
rule_53: A30 -> 2
rule_54: A31 -> 1
rule_55: A31 -> 2
rule_56: A31 -> 20
rule_57: A32 -> 1
rule_58: A32 -> 2
rule_59: A33 -> A34
rule_60: A33 -> A35
rule_61: A33 -> A36
rule_62: A33 -> A37
rule_63: A33 -> A38
rule_64: A33 -> A39
rule_65: A33 -> A40
rule_66: A33 -> A41
rule_67: A33 -> A42
rule_68: A33 -> A43
rule_69: A33 -> A44
rule_70: A33 -> A45
rule_71: A34 -> 6
rule_72: A35 -> 6
rule_73: A36 -> 6
rule_74: A37 -> 6
rule_75: A38 -> 6
rule_76: A39 -> 6
rule_77: A40 -> 6
rule_78: A41 -> 6
rule_79: A42 -> 6
rule_80: A43 -> 6
rule_81: A44 -> 6
rule_82: A45 -> 6
rule_83: A46 -> 21 A47
rule_84: A47 -> 7
rule_85: A48 -> A49 A46 A18 A33
rule_86: A49 -> 22 A24
rule_87: A50 -> 23
rule_88: A50 -> A51 A29
rule_89: A51 -> 24
rule_90: A52 -> A53 A25
rule_91: A53 -> 25
rule_92: A54 -> A57 A59 A13 A58
rule_93: A55 -> 27 A52 28
rule_94: A56 -> 29 A50
rule_95: A57 -> 31 A8
rule_96: A58 -> 32 A50 33
rule_97: A59 -> 34 A48 35
rule_98: A60 -> A54 37 A61 39
rule_99: A61 -> A62 A56 A3 A55
rule_100: A62 -> 40 A7
rule_101: A63 -> 41 A60

Target sequence:
41 31 11 3 1 34 22 18 21 7 17 16 2 6 35 12 4 13 32 24 15 20 33 26 37 40 10 29 23 30 8 5 9 27 25 14 19 28 39 36 38
"""